unstable periodic orbits

In a mathematical space whose coordinates represent the state of a dynamical system (i.e., a state space), periodic orbits are the set of equilibrium states. If all of the periodic orbits in this abstract dynamical landscape are unstable, the system's temporal evolution will never settle down to any one of them. Instead, system's behavior wanders incessantly in a sequence of close approaches to these orbits. The more unstable an orbit, the less time the system spends near it. Unstable Periodic Orbits (UPOs) form the "skeleton" of nonlinear dynamics, and one can build a model of a system by counting and characterizing its UPOs in a hierarchy of orbits with increasing periodicity. Model's accuracy can be improved by progressivelly adding longer-period orbits to the hierarchy. The dynamical landscape can then be tesselated into regions of the state space centered around these UPOs. Orbit locations and stability can also offer short-term predictions for the system's future states. This type of predictive model can be used for parametric control of nonlinear systems, whether they are chaotic or not. However, rigorous identification of UPOs from noisy experimental data is a difficult task.

There's a straightforward method for identification of UPOs which relies on the recurrence of patterns in state space, though that's a very rare event (i.e., a state repeatedly returning near an orbit) in the reality of short and nonstationary datasets. There's another method based on a local dynamics data transformation, which acts as a dynamical lens so that the new datasets are concentrated about distinct UPOs, helping to offset the usual scarcity of trajectories near UPOs. With the additional ability to identify complex higher period orbits by using only fragments of trajectories near those orbits, identification of UPOs was successfully achieved in various experimental settings, including epileptiform activities from the human cortex.

Tracking "parameter" changes from the inherently nonstationary data of, e.g., neurological systems, with UPOs has also been accomplished, as this is a strong requirement for UPO-based control of nonlinear systems. Furthermore, this tracking could be used to detect changes in system state due to intrinsic "parameter" variations (e.g., the transition to epileptic seizures), extrinsic effects (e.g., due to electromagnetic fields), or even perceptual discrimination.